Browsing by Author "Shahmorad, Sedaghat"
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Article Existence, Uniqueness and Blow-Up of Solutions for Generalized Auto-Convolution Volterra Integral Equations(Elsevier Science inc, 2024) Mostafazadeh, Mahdi; Shahmorad, Sedaghat; Erdogan, FevziIn this paper, our intention is to investigate the blow-up theory for generalized auto -convolution Volterra integral equations (AVIEs). To accomplish this, we will consider certain conditions on the main equation. This will establish a framework for our analysis, ensuring that the solution of the equation exists uniquely and is positive. Firstly, we analyze the existence and uniqueness of a local solution for a more general class of AVIEs (including the proposed equation in this paper) under certain hypotheses. Subsequently, we demonstrate the conditions under which this local solution blows up at a finite time. In other words, the solution becomes unbounded at that time. Furthermore, we establish that this blow-up solution can be extended to an arbitrary interval on the non -negative real line, thus referred to as a global solution. These results are also discussed for a special case of generalized AVIEs in which the kernel functions are taken as positive constants.Article Review of Recursive and Operational Approaches of the Tau Method With a New Extension(Springer Heidelberg, 2023) Shahmorad, Sedaghat; Talaei, Younes; Tunc, CemilThis is a review paper that briefly represents the recursive and operational approaches to the Tau method on solving ordinary differential and integro-differential equations with suitable initial or boundary conditions, and we discuss a new extension of the method on solving a class of Abel Volterra integral equations which can be also used for solving fractional differential equations. Extension of height and canonical polynomials are introduced. Illustrative examples are given in each case to clarify the performance and structural properties of the method.